Advanced search

Advanced search is divided into two main parts, and one or more groups in each of the main parts. The main parts are the "Search for" (including) and the "Remove from search" (excluding) part. (The excluding part might not be visible until you hit "NOT" for the first time.) You can add new groups to both the including and the excluding part by using the buttons "OR" or "NOT" respectively, and you can add more search options to all groups through the drop down menu on the last row (in each group).

For a result to be included in the search result, is it required to fit all added including parameters (in at least one group) and not fit all parameters in one of the excluding groups. This system with the two main parts and their groups makes it possible to combine two (or more) distinct searches into one search result, while being flexible in removing results from the final list.

When acts on the flag variety of , the orbits are in bijection with fixed point free involutions in the symmetric group . In this case, the associated Kazhdan-Lusztig-Vogan polynomials can be indexed by pairs of fixed point free involutions , where denotes the Bruhat order on . We prove that these polynomials are combinatorial invariants in the sense that if is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then for all v amp;gt;= u.

In this thesis the relationship between Gröbner bases and algebraic coding theory is investigated, and especially applications towards linear codes, with Reed-Müller codes as an illustrative example. We prove that each linear code can be described as a binomial ideal of a polynomial ring, and that a systematic encoding algorithm for such codes is given by the remainder of the information word computed with respect to the reduced Gröbner basis. Finally we show how to apply the representation of a code by its corresponding polynomial ring ideal to construct a class of codes containing the so called primitive Reed-Müller codes, with a few examples of this result.

We study various boundary and inner regularity questions for p(.)-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for p(.)-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded p(.)-harmonic functions and give some new characterizations of W-0(1,p(.)) spaces. We also show that p(.)-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

Summary. Questions of existence and uniqueness for discrete frictional quasi-static incremental problems, rate problems and wedging problems are discussed. Various methods to compute critical bounds for the coefficient of friction which guarantee existence and uniqueness are described, as well as the sharpness of the bounds and their interdependence.

We suggest a method of constructing decompositions of a topological space X having an open subset homeomorphic to the space (R-n , tau), where n is an integer greater than= 1 and tau is any admissible extension of the Euclidean topology of R-n (in particular, X can be a finite-dimensional separable metrizable manifold), into a countable family F of sets (dense in X and zero-dimensional in the case of manifolds) such that the union of each non-empty proper subfamily of F does not have the Baire property in X.

We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each n greater than 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of R" without the Baire property.

The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R-n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

In this paper, we present a qualitative study in which we analyse the video-recordings of four groups of students solving Fermi Problems. Previous studies show that Secondary School students solve this type of problems using complex problem solving processes and developing mathematical models. In order to analyse the students’ problem solving processes, so-called Modelling Activity Diagrams were used. The results of the present study demonstrate that solving Fermi problems is a complex matter, and that some of the theoretical tools used in the field of Mathematical Education fail to adequately reflect this level of complexity. In addition, Modelling Activity Diagrams are presented as a more detailed analysis tool to characterise student choices and actions, as well as to make the structure of the Fermi problem addressed more visible.

Many students of secondary school find it difficult to understand algebra, specifically linear equations and systems of linear equations, which prevents their further learning in both mathematics and other subjects. Teacher must therefore have a clear idea of what these difficulties are in order to assist their students in understanding and learning this area in algebra. The study examines the findings of the research literature as to what these difficulties are. Proven difficulties that have been illustrated with examples of students´ solutions are: deficiencies in their algebraic prerequisites, procedural knowledge and conceptual knowledge (especially about similarities, constants and variables), lack of knowledge of the algebraic syntax, and incorrect handling of the operations used to solve equations and systems of equations, for example how to handle negative coefficients and constants and use the substitution method.

The focus of this thesis is to introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds. We first give here a review of the geometry of Kähler manifolds and Lie-Rinehart algebras. After that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts. Furthermore, we introduce a concept of morphism between Kähler-Poisson algebras and show its consequences. Detailed examples are provided in order to illustrate the novel concepts.

The selection of a mine design is based on estimating net present values of all possible, technically feasible mine plans so as to select the one with the maximum value. It is a hard task to know with certainty the quantity and quality of ore in the ground. This geological uncertainty and also the future market behavior of metal prices and foreign exchange rates, which are always uncertain, make mining a high risk business. Value-at-Risk (VaR) is a measure that is used in financial decisions to minimize the loss caused by inadequate monitoring of risk. This measure does, however, have certain drawbacks such as lack of consistency, nonconvexity, and nondifferentiability. Rockafellar and Uryasev [J. Risk 2, 21-41 (2000)] introduce the Conditional Value-at-Risk (CVaR) measure as an alternative to the VaR measure. The CVaR measure gives rise to a convex optimization problem. An optimization model that maximizes expected return while minimizing risk is important for the mining sector as this will help make better decisions on the blocks of ore to mine at a particular point in time. We present a CVaR approach to the uncertainty involved in open-pit mining. We formulate investment and design models for the open-pit mine and also give a nested pit scheduling model based on CVaR. Several numerical results based on our models are presented by using scenarios from simulated geological and market uncertainties.

The purpose of this study is to investigate homework given in the flipped mathematics classroom. One of the characteristics of the “flipped classroom” is that traditional lectures are not placed in class time. Direct instruction is instead given as homework, “flipped homework”, often in the form of video lectures. The literature review is based on nine articles and focuses on the design of flipped mathematics homework, pupil’s views of the method, and the possible advantages and disadvantages of flipped homework in relation to traditional homework. There is still a lack of research done on “flipped homework”, which makes it difficult to draw any general conclusions. However, the results indicate that the teaching method may have some advantages, including that video lectures gives the students a greater responsibility for their own learning, and that the fixed time of the video have the potential to reduce the difference in time spent by different students on the same homework.

Human activity and other events can cause environmental changes to the habitat of organisms. The environmental changes effect the vital rates for a population. In order to predict the impact of these environmental changes on populations, we use two different models for population dynamics. One simpler linear model that ignores environmental competition between individuals and another model that does not. Our population models take into consideration the age distribution of the population and thus takes into consideration the impact of demographics. This thesis generalize two theorems, one for each model, developed by Sonja Radosavljevic regarding long term upper and lower bounds of a population with periodic birth rate ; see [6] and [5]. The generalisation consist in including the case where the periodic part of the birth rate can be expressed with a finite Fourier series and also infinite Fourier series under some constraints. The old theorems only considers the case when the periodic part of the birth rate can be expressed with one cosine term. From the theorems we discover a connection between the frequency of oscillation and the effect on population growth. From this derived connection we conclude that periodical changing environments can have both positive and negative effects on the population.

The Encyclopedia of Thermal Stresses is an important interdisciplinary reference work. In addition to topics on thermal stresses, it contains entries on related topics, such as the theory of elasticity, heat conduction, thermodynamics, appropriate topics on applied mathematics, and topics on numerical methods. The Encyclopedia is aimed at undergraduate and graduate students, researchers and engineers. It brings togetherwell established knowledge and recently received results. All entries were prepared by leading experts from all over the world, and are presented in an easily accessible format. The work is lavishly illustrated, examples and applications are given where appropriate, ideas for further development abound, and the work will challenge many students and researchers to pursue new results of their own. This work can also serve as a one-stop resource for all who need succinct, concise, reliable and up to date information in short encyclopedic entries, while the extensive references will be of interest to those who need further information. For the coming decade, this is likely to remain the most extensive and authoritative work on Thermal Stresses

This paper explores the effect of initial conditions on the behavior of coupled frictional elastic systems subject to periodic loading. Previously, it has been conjectured that the long term response will be independent of initial conditions if all nodes slip at least once during each loading cycle. Here, this conjecture is disproved in the context of a simple two-node system. Counter examples are presented of “unstable” steady-state orbits that repel orbits starting from initial conditions that are sufficiently close to the steady state. The conditions guaranteeing stability of such steady states are shown to be more restrictive than those required for the rate problem to be uniquely solvable for arbitrary derivative of the external loading. In cases of instability, the transient orbit is eventually limited either by slip occurring at both nodes simultaneously, or by one node separating. In both cases a stable limit cycle is obtained. Depending on the slopes of the constraint lines, the limit cycle can involve two periods of the loading cycle, in which case it appears to be unique, or it may repeat every loading cycle, in which case distinct limit cycles are reached depending on the sign of the initial deviation from the steady state. In the case of instability an example is given of a loading for which a quasi-static evolution problem with multiple solutions exists, whereas all rate problems are uniquely solvable.

We consider the class of two or three-dimensional discrete contact problems in which a set of contact nodes can make frictional contact with a corresponding set of rigid obstacles. Such a system might result from a finite element discretization of an elastic contact problem after the application of standard static reduction operations. The Coulomb friction law requires that the tractions at any point on the contact boundary must lie within or on the surface of a friction cone, but the exact position of any stuck node (i.e., a node where the tractions are strictly within the cone) depends on the initial conditions and/or the previous history of loading. If the long-term loading is periodic in time, we anticipate that the system will eventually approach a steady periodic cycle. Here we prove that if the elastic system is uncoupled, meaning that changes in slip displacements alone have no effect on the instantaneous normal contact reactions, the time-varying terms in this steady cycle are independent of initial conditions. In particular, we establish the existence of a unique permanent stick zone T comprising the set of all nodes that do not slip after some finite number of cycles. We also prove that the tractions and slip velocities at all nodes not contained in T approach unique periodic functions of time, whereas the (time-invariant) slip displacements in T may depend on initial conditions. Typical examples of uncoupled systems include those where the contact surface is a plane of symmetry, or where the contacting bodies can be approximated locally as half spaces and Dundurs mismatch parameter beta = 0. An important consequence of these results is that systems of this kind will exhibit damping characteristics that are independent of initial conditions. Also, the energy dissipated at each slipping node in the steady state is independent of initial conditions, so wear patterns and the incidence of fretting fatigue failure should also be so independent.

We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Subdivision surfaces permit a designer to specify the approximate form of a surface defining an object and to refine and smooth the form to obtain a more useful or attractive version of the surface.

A considerable amount of mathematical theory is required to understand the characteristics of the resulting surfaces, and this book

• provides a careful and rigorous presentation of the mathematics underlying subdivision surfaces as used in computer graphics and animation, explaining the concepts necessary to easily read the subdivision literature;

• organizes subdivision methods into a unique and unambiguous hierarchy to facilitate insight and understanding;

• gives a broad discussion of the various methods and is not restricted to questions related to regularity of subdivision surfaces at so-called extraordinary points.

Introduction to the Mathematics of Subdivision Surfaces is excellent preparation for reading more advanced texts that delve more deeply into special questions of regularity. The authors provide exercises and projects at the end of each chapter. Course material, including solutions to the exercises, is available on an associated Web page.

The multilinear least-squares (MLLS) problem is an extension of the linear leastsquares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by results of numerical experiments performed for some problems related to the design of filter networks.

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m andlt;= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -andgt; infinity.

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature.

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on ( ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. (C) 2015 Elsevier Ltd. All rights reserved.

We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the *-algebras can be completed into C*-algebras isomorphic to the standard non-commutative torus. As the former was constructed in the context of matrix (or fuzzy) geometries, it provides an important link to the framework of non-commutative geometry, and opens up for a concrete way to study deformations of non-commutative tori. Furthermore, we show how the well-known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite-dimensional representations of the deformed algebras, and their projective modules are described together with connections of constant curvature.

A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex structure, and the curvature is explicitly calculated. A noncommutative analogue of the fact that the catenoid is a minimal surface is studied by constructing a Laplace operator from the connection and showing that the embedding coordinates are harmonic. Furthermore, an integral is defined and the total curvature is computed. Finally, classes of left and right modules are introduced together with constant curvature connections, and bimodule compatibility conditions are discussed in detail.

It is pointed out that the equations less thanbrgreater than less thanbrgreater thanSigma(d)(i=1)[X-i, [X-i, X-j]] = 0 less thanbrgreater than less thanbrgreater than(and its super-symmetrizations, playing a central role in M-theory matrix models) describe non-commutative minimal surfaces - and can be solved as such.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingartens formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative Gauss-Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kahler geometry.

We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Chern-Gauss-Bonnet type theorem for the noncommutative 4-sphere. (C) 2016 Elsevier B.V. All rights reserved.

In order to investigate to what extent the calculus of classical (pseudo-) Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civitas theorem, which states that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.

Consider a company which produces and sells a certain product on a market with highly variable demand. Since the demand is very high during some periods, the company will produce and create a stock in advance before these periods. On the other hand it costs money to hold a big stock, so that some balance is needed for optimum. The demand is assumed to be known in advance with sufficient accuracy. We use a technique from optimal control theory for the analysis, which leads to so-called activity periods. During such a period the stock is positive and the production is maximal, provided that the problem starts with zero stock, which is the usual case. Over a period of one or more years, there will be a few activity periods. Outside these periods the stock is zero and the policy is to choose production = the smaller of [demand, maximal production]. The “intrinsic time length” is a central concept. It is simply the maximal time a unit of the product can be stored before selling without creating a loss.

Consider a company which produces and sells a certain product on a market with highly variable demand. Since the demand is very high during some periods, the company will produce and create a stock in advance before these periods. On the other hand it costs money to hold a big stock, so that some balance is needed for optimum. The demand is assumed to be known in advance with sufficient accuracy. We use a technique from optimal control theory for the analysis, which leads to so-called activity periods. During such a period the stock is positive and the production is maximal, provided that the problem starts with zero stock, which is the usual case. Over a period of one or more years, there will be a few activity periods. Outside these periods the stock is zero and the policy is to choose production = the smaller of [demand, maximal production]. The “intrinsic time length” is a central concept. It is simply the maximal time a unit of the product can be stored before selling without creating a loss.

Various approaches are used to derive the Aronsson-Euler equations for L-infinity calculus of variations problems with constraints. The problems considered involve holonomic, nonholonomic, isoperimetric, and isosupremic constraints on the minimizer. In addition, we derive the Aronsson-Euler equation for the basic L-infinity problem with a running cost and then consider properties of an absolute minimizer. Many open problems are introduced for further study.

We claim that one of the authors of the article On universal groups and three-manifolds, Invent. Math. 87 (1987), no. 3, 441–456 is W. Witten. The correct name of this author is Wilbur Carrington Whitten. W. C. Whitten is not the father of the fields medallist E. Witten, as we claim there.

María Teresa, Maite, Lozano is a great person and mathematician, in these pages we can only give a very small account of her results trying to resemble her personality. We will focus our attention only on a few of the facets of her work, mainly in collaboration with Mike Hilden and José María Montesinos because as Maite Lozano pointed out in an international conference in Umeå University in June 2017, where she was a plenary speaker:

I am specially proud of been part of the team Hilden-Lozano-Montesinos (H-L-M), and of our mathematical achievements

Let (Y0,Y1) be a Banach couple and let Xj be a closed complemented subspace of Yj, (j = 0,1). We present several results for the general problem of finding necessary and sufficient conditions on the parameters (θ, q) such that the real interpolation space (X0,X1)θ,q is a closed subspace of (Y0,Y1)θ,q. In particular, we establish conditions which are necessary and sufficient for the equality (X0,X1)θ,q = (Y0,Y1)θ,q, with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that X1 = Y1 and X0 is a subspace of codimension one in Y0.

For p is an element of [1, infinity], we establish criteria for the one-sided invertibility of binomial discrete difference operators A = aI - bV on the space l(p) = l(p)(Z), where a, b is an element of l(infinity), I is the identity operator and the isometric shift operator V is given on functions f. lp by (Vf)(n) = f (n+ 1) for all n is an element of Z. Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators A = aI - bU(alpha) on the Lebesgue space L-p(R+) for every p is an element of [1, infinity], where a, b is an element of L-infinity (R+), a is an orientation-preserving bi-Lipschitz homeomorphism of [0, +infinity] onto itself with only two fixed points 0 and infinity, and U-alpha is the isometric weighted shift operator on L-p(R+) given by U(alpha)f = (alpha)(1/p)(f circle alpha). Applications of binomial discrete operators to interpolation theory are given.

Let A be a bounded linear operator from a couple (X-0, X-1) to a couple (Y-0, Y-1) such that the restrictions of A on the end spaces X-0 and X-1 have bounded inverses defined on Y-0 and Y-1, respectively. We are interested in the problem of how to determine if the restriction of A on the space (X-0, XI)(theta,q) has a bounded inverse defined on the space (Y-0, Y-1)(theta,q). In this paper, we show that a solution to this problem can be given in terms of indices of two subspaces of the kernel of the operator A on the space X-0 + X-1.

Findings from interviews investigating how Swedish first year engineering students recognize undergraduate mathematics texts as being more or less “mathematical”. The results indicate a relation between the students’ understanding of the principles for knowledge classification and their success in their mathematics studies.

Let G be a Class 1 graph with maximum degree 4 and let t amp;gt;= 5 be an integer. We show that any proper t-edge coloring of G can be transformed to any proper 4-edge coloring of G using only transformations on 2-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper 5-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent. (C) 2015 Wiley Periodicals, Inc.